Question: The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives $22.6$ years; the standard deviation is $5.3$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a gorilla living less than $12$ years.
$22.6$ $17.3$ $27.9$ $12$ $33.2$ $6.7$ $38.5$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $22.6$ years. We know the standard deviation is $5.3$ years, so one standard deviation below the mean is $17.3$ years and one standard deviation above the mean is $27.9$ years. Two standard deviations below the mean is $12$ years and two standard deviations above the mean is $33.2$ years. Three standard deviations below the mean is $6.7$ years and three standard deviations above the mean is $38.5$ years. We are interested in the probability of a gorilla living less than $12$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the gorillas will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the gorillas will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $12$ years and the other half $({2.5\%})$ will live longer than $33.2$ years. The probability of a particular gorilla living less than $12$ years is ${2.5\%}$.